18.S190 | January IAP 2023 | Undergraduate

# Introduction to Metric Spaces

## Lecture Notes

#### Lecture 1: Motivation, Intuition, and Examples

Outline: Motivation, definition, and intuition behind metric spaces. Redefining 18.100A Real Analysis and 18.100P Real Analysis in terms of metrics: open/closed sets, convergence, Cauchy sequences, and continuity.

Lecture 1: Motivation, Intuition, and Examples (PDF)

#### Lecture 2: General Theory

Outline: Some general theory of metric spaces regarding convergence, open and closed sets, continuity, and their relationship to one another.

Lecture 2: General Theory (PDF)

Lecture 2: General Theory (TEX)

#### Lecture 3: Compact Sets in Rⁿ

Outline: Norms and analysis on finite sets (as motivation for compact sets). Topological compactness and sequential compactness. Today will focus purely on theorems regarding compact subsets of Rⁿ. The Heine-Borel theorem and the Bolzano-Weierstrass theorem.

Lecture 3: Compact Sets in Rⁿ (PDF)

Lecture 3: Compact Sets in Rⁿ (TEX)

#### Lecture 4: Compact Metric Spaces

Outline: Compact sets on general metric spaces. Showing sequential compactness is equivalent to topological compactness, which is equivalent to being totally bounded and complete (on metric spaces).

Lecture 4: Compact Metric Spaces (PDF)

#### Lecture 5: Complete Metric Spaces

Outline: Completions of metric spaces, motivating Lp spaces, Sobolev spaces, p-adic numbers, Banach spaces, and Hilbert spaces.

Lecture 5: Complete Metric Spaces (PDF)

Lecture 5: Complete Metric Spaces (TEX)

#### Lecture 6: Where We Go from Here

Outline: How these topics were motivated, and a preview of how these topics come up in later classes.

Lecture 6: Where We Go from Here (PDF)

Lecture 6: Where We Go from Here (TEX)

## Course Info

January IAP 2023
##### Learning Resource Types
Lecture Notes
Lecture Videos
Online Textbook
Problem Sets
Instructor Insights